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1. ONe Index for All Kernels (ONIAK): A Zero Re-Indexing LSH Solution to ANNS-ALT (After Linear Transformation)

   https://doi.org/10.14778/3565838.3565847  

   Meng, Jingfan; Wang, Huayi; Xu, Jun; Ogihara, Mitsunori (September 2022, Proceedings of the VLDB Endowment)

   In this work, we formulate and solve a new type of approximate nearest neighbor search (ANNS) problems called ANNS after linear transformation (ALT). In ANNS-ALT, we search for the vector (in a dataset) that, after being linearly transformed by a user-specified query matrix, is closest to a query vector. It is a very general mother problem in the sense that a wide range of baby ANNS problems that have important applications in databases and machine learning can be reduced to and solved as ANNS-ALT, or its dual that we call ANNS-ALTD. We propose a novel and computationally efficient solution, called ONe Index for All Kernels (ONIAK), to ANNS-ALT and all its baby problems when the data dimension d is not too large (say d ≤ 200). In ONIAK, a universal index is built, once and for all, for answering all future ANNS-ALT queries that can have distinct query matrices. We show by experiments that, when d is not too large, ONIAK has better query performance than linear scan on the mother problem (of ANNS-ALT), and has query performances comparable to those of the state-of-the-art solutions on the baby problems. However, the algorithmic technique behind this universal index approach suffers from a so-called dimension blowup problem that can make the indexing time prohibitively long for a large dataset. We propose a novel algorithmic technique, called fast GOE quadratic form (FGoeQF), that completely solves the (prohibitively long indexing time) fallout of the dimension blowup problem. We also propose a Johnson-Lindenstrauss transform (JLT) based ANNS-ALT (and ANNS-ALTD) solution that significantly outperforms any competitor when d is large. 

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2. SeRF: Segment Graph for Range-Filtering Approximate Nearest Neighbor Search

   https://doi.org/10.1145/3639324  

   Zuo, Chaoji; Qiao, Miao; Zhou, Wenchao; Li, Feifei; Deng, Dong (March 2024, Proceedings of the ACM on Management of Data)

   Effective vector representation models, e.g., word2vec and node2vec, embed real-world objects such as images and documents in high dimensional vector space. In the meanwhile, the objects are often associated with attributes such as timestamps and prices. Many scenarios need to jointly query the vector representations of the objects together with their attributes. These queries can be formalized as range-filtering approximate nearest neighbor search (ANNS) queries. Specifically, given a collection of data vectors, each associated with an attribute value whose domain has a total order. The range-filtering ANNS consists of a query range and a query vector. It finds the approximate nearest neighbors of the query vector among all the data vectors whose attribute values fall in the query range. Existing approaches suffer from a rapidly degrading query performance when the query range width shifts. The query performance can be optimized by a solution that builds an ANNS index for every possible query range; however, the index time and index size become prohibitive -- the number of query ranges is quadratic to the number n of data vectors. To overcome these challenges, for the query range contains all attribute values smaller than a user-provided threshold, we design a structure called the segment graph whose index time and size are the same as a single ANNS index, yet can losslessly compress the n ANNS indexes, reducing the indexing cost by a factor of Ω(n). To handle general range queries, we propose a 2D segment graph with average-case index size O(n log n) to compress n segment graphs, breaking the quadratic barrier. Extensive experiments conducted on real-world datasets show that our proposed structures outperformed existing methods significantly; our index also exhibits superior scalability. 

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3. Optimal Dynamic Regret in Proper Online Learning with Strongly Convex Losses and Beyond

   Baby, Dheeraj; Wang, Yu-Xiang (April 2022, Proceedings of Machine Learning Research)

   We study the framework of universal dynamic regret minimization with strongly convex losses. We answer an open problem in Baby and Wang 2021 by showing that in a proper learning setup, Strongly Adaptive algorithms can achieve the near optimal dynamic regret of 𝑂̃ (𝑑1/3𝑛1/3TV[𝑢1:𝑛]2/3∨𝑑) against any comparator sequence 𝑢1,…,𝑢𝑛 simultaneously, where 𝑛 is the time horizon and TV[𝑢1:𝑛] is the Total Variation of comparator. These results are facilitated by exploiting a number of new structures imposed by the KKT conditions that were not considered in Baby and Wang 2021 which also lead to other improvements over their results such as: (a) handling non-smooth losses and (b) improving the dimension dependence on regret. Further, we also derive near optimal dynamic regret rates for the special case of proper online learning with exp-concave losses and an 𝐿∞ constrained decision set. 

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4. ARKGraph: All-Range Approximate K-Nearest-Neighbor Graph

   https://doi.org/10.14778/3603581.3603601  

   Zuo, Chaoji; Deng, Dong (June 2023, Proceedings of the VLDB Endowment)

   Given a collection of vectors, the approximate K-nearest-neighbor graph (KGraph for short) connects every vector to its approximate K-nearest-neighbors (KNN for short). KGraph plays an important role in high dimensional data visualization, semantic search, manifold learning, and machine learning. The vectors are typically vector representations of real-world objects (e.g., images and documents), which often come with a few structured attributes, such as times-tamps and locations. In this paper, we study the all-range approximate K-nearest-neighbor graph (ARKGraph) problem. Specifically, given a collection of vectors, each associated with a numerical search key (e.g., a timestamp), we aim to build an index that takes a search key range as the query and returns the KGraph of vectors whose search keys are within the query range. ARKGraph can facilitate interactive high dimensional data visualization, data mining, etc. A key challenge of this problem is the huge index size. This is because, given n vectors, a brute-force index stores a KGraph for every search key range, which results in O (K n 3 ) index size as there are O ( n 2 ) search key ranges and each KGraph takes O (K n ) space. We observe that the KNN of a vector in nearby ranges are often the same, which can be grouped together to save space. Based on this observation, we propose a series of novel techniques that reduce the index size significantly to just O (K n log n ) in the average case. Furthermore, we develop an efficient indexing algorithm that constructs the optimized ARKGraph index directly without exhaustively calculating the distance between every pair of vectors. To process a query, for each vector in the query range, we only need O (log log n + K log K) to restore its KNN in the query range from the optimized ARKGraph index. We conducted extensive experiments on real-world datasets. Experimental results show that our optimized ARKGraph index achieved a small index size, low query latency, and good scalability. Specifically, our approach was 1000x faster than the baseline method that builds a KGraph for all the vectors in the query range on-the-fly. 

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5. Improved Complexity for Smooth Nonconvex Optimization: A Two-Level Online Learning Approach with Quasi-Newton Methods

   https://doi.org/10.1145/3717823.3718308  

   Jiang, Ruichen; Mokhtari, Aryan; Patitucci, Francisco (June 2025, ACM)

   We study the problem of finding an 𝜀-first-order stationary point (FOSP) of a smooth function, given access only to gradient information. The best-known gradient query complexity for this task, assuming both the gradient and Hessian of the objective function are Lipschitz continuous, is O(𝜀−7/4). In this work, we propose a method with a gradient complexity of O(𝑑1/4𝜀−13/8), where 𝑑 is the problem dimension, leading to an improved complexity when 𝑑 = O(𝜀−1/2). To achieve this result, we design an optimization algorithm that, underneath, involves solving two online learning problems. Specifically, we first reformulate the task of finding a stationary point for a nonconvex problem as minimizing the regret in an online convex optimization problem, where the loss is determined by the gradient of the objective function. Then, we introduce a novel optimistic quasi-Newton method to solve this online learning problem, with the Hessian approximation update itself framed as an online learning problem in the space of matrices. Beyond improving the complexity bound for achieving an 𝜀-FOSP using a gradient oracle, our result provides the first guarantee suggesting that quasi-Newton methods can potentially outperform gradient descent-type methods in nonconvex settings. 

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